Strong amalgamation of distributive lattices

Abstract

The class D of distributive lattices is known to have the Amalgamation Property: If A and B are distributive lattices, and S is a common sublattice of A and B, then there exists a distributive lattice L containing A and B as sublattices sharing S as a sublattice. However, in general, we cannot guarantee that L can be chosen so that A ( B = S holds in L; in other words, the Strong Amalgamation Property fails in D. As a trivial example (see Fig.!), let A={sO,a,sl,s2} be the fourelement Boolean lattice with So as the zero and S2 as the unit; similarly, B = {so, b, s l' S2 }. Then S= A ( B = {so, Sl' S2} is the three-element chain. Now, if L is a distributive lattice containing A and Bas sublattices, A and B sharing S as a sublattice, then in L both a and b are relative complements of SI in the interval [so, S2]; hence, by the distributivity of L, a = b in L. Thus A ( B contains a( = b), and thus A ( B = S fails in L. Our main result states that this trivial example is typical. Let A and B be lattices, and let S be a common sublattice of A and B. A partial M 3 contained in A and B over S (in short, in A, B, S) consists of five elements So, Sl' S2' a, b such that {so, Sl' S2} is a three-element chain in Sand {so,a,s.,s2} (resp., {SO,b,Sl,S2}) is a sublattice of A (resp., of B) isomorphic to the four-element Boolean lattice with So as zero and S2 as unit (see Fig. 2).